But it is evident from the form of these equations, that we may satisfy the whole system by making

B1 = B. a, BB1.a', B1 = B1. a, B1 = B. a, &c.

provided we determine B, by

ƒ (x', 'y', z') = ƒ'® = B ̧+ B ̧r" + B,ƒa + &c. = B, (1 − ~,~)~,;

In the value of p just exhibited, the radius of the sphere is taken as the unit of space, but the same formula may easily be

a

adapted to any other unit by writing and in the place of Ђ Ђ

a and r' respectively, and recollecting at the same time that in consequence of the equation

before given,, is a quantity of the dimension -1 with regard

b

to space: being the number which represents the radius of the sphere when we employ the new unit. Hence we obtain for a sphere whose radius is bg, acted upon by an exterior concentric spherical surface of which the radius is a,

P' being the density of the fluid on the exterior surface.

If now we conceive a conducting sphere A whose radius is a, and determine P' so that all the fluid of one kind, viz. that which is redundant in this sphere, may be condensed on its surface, and afterwards find b the radius of the interior sphere B from the condition that it shall just contain all the fluid of the opposite kind, it is evident that each of the fluids will be in equilibrium within A, and therefore the problem originally proposed is thus accurately solved. The reason for supposing all the fluid of one name to be completely abstracted from B, is that our formulæ may represent the state of permanent equilibrium, for the tendency of the forces acting within the void shell included between the surfaces A and B, is to abstract continually the fluid of the same name as that on A's surface from the sphere B.

To prove the truth of what has just been asserted, we will begin with determining the repulsion exerted by the inner sphere itself, on any point p exterior to it, and situate at the distance? from its centre O. But by what Laplace has shown (Mec. Cel. Liv. II. No. 12) the repulsion on an exterior point p, arising from a spherical shell of which the radius is r', thickness dr' and centre is at O will be measured by

and the part of the required repulsion due thereto will, by substituting for p its value before found, become

It now remains to find the value of the definite integral herein

contained.

But when (1-2) is expanded, and the integra

But when

where after the integrations have been effected, x ought to be

The value of the integral last found being substituted in the expression immediately preceding, and the finite integral taken relative to s from 80 to 8∞o gives for the repulsion of the inner sphere,

But we have evidently by means of the binomial theorem,

If now we make x= r, the same quantity may be written

Having thus the value of the repulsion due to the inner sphere B on an exterior point p, it remains to determine that due to the fluid on A's surface. But this last is represented by

2παρ' d. (a + r)3* — (a — g) 3* 1-n.3-n dr

..(17).

(Mec. Cel. Liv. IL No. 12.) Now by expanding this function

there results

The last of these expressions may readily be exhibited under

a finite form, by remarking that

Sox"da (1 − x") "*" ( 1 − 12x2) **

= [','x ̃dx (1 − x2) == ≤ n . n + 2 ...... n + 28 − 2 pole pole

2.4.6....

28

n.n+1.n+2...n+28-1

2

Σε

3

4. 5 6

28+3

(8+1).

Hence, since T(), the value of the repulsion arising from A's surface becomes

Now by adding the repulsion due to the inner sphere which is given by the formula (16), we obtain, (since it is evidently indifferent what variable enters into a definite integral, provided each of its limits remain unchanged)